If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq x\}$.

Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$

The *Wallman topology* on $\text{NPU}(\omega)$ is defined as follows: For $A\subseteq \omega$ set $\Phi_A = \{{\cal U} \in \text{NPU}(\omega): A\in{\cal U}\}$. Then $${\frak S} = \big\{\text{NPU}(\omega) \setminus \Phi_A: A\subseteq \omega\big\}$$ is a subbasis for the Wallman topology on $\text{NPU}(\omega)$.

**Question.** Pick any ${\cal U}\in \text{NPU}(\omega)$. Are $(\downarrow_{\leq_{RK}} {\cal U})$ and $(\uparrow_{\leq_{RK}} {\cal U})$ closed in the Wallman topology on $\text{NPU}(\omega)$?

(Note. This would imply that the Wallman topology contains the interval topology of $(\text{NPU}(\omega), \leq_{RK})$.)